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A299144 a(n) is the least i such that gcd(Fibonacci(i), i+x) > 1 for all x=0..n.

%I A299144 #30 Nov 24 2024 20:15:27
%S A299144 5,10,18,30,30,30,30,180,180,180,180,840,840,1260,1260,1260,1260,
%T A299144 24480,24480,63000,63000,63000,63000,63000,63000,63000,63000,63000,
%U A299144 63000,356400,356400,356400,356400,356400,356400,356400,356400,5783400,5783400,5783400,5783400,5783400,5783400
%N A299144 a(n) is the least i such that gcd(Fibonacci(i), i+x) > 1 for all x=0..n.
%e A299144 5 is the smallest integer i such that gcd(F(i), i) > 1, because F(5)=5. Therefore a(0)=5.
%e A299144 10 is the smallest integer i such that gcd(F(i), i) > 1 and gcd(F(i), i+1) > 1, because F(10)=55, not coprime to 10 nor 11. Therefore a(1)=10.
%t A299144 Nest[Function[a, Append[a, SelectFirst[Range[10^5], Function[i, AllTrue[i + Range[0, Length@ a], ! CoprimeQ[Fibonacci@ i, #] &]]]]], {}, 29] (* _Michael De Vlieger_, Feb 05 2018 *)
%o A299144 (Python)
%o A299144 p0=0
%o A299144 p1=1
%o A299144 def GCD(x,y):
%o A299144     tmp = y
%o A299144     y = x % y
%o A299144     if y==0: return tmp
%o A299144     return GCD(tmp, y)
%o A299144 n=0
%o A299144 for i in range(1,1000000):
%o A299144     p0,p1 = p1, p0+p1
%o A299144     for x in range(1000000):
%o A299144         if GCD(p0,i+x)==1: break
%o A299144     for j in range(n, x):
%o A299144         print(i)
%o A299144     if x>n: n=x
%o A299144 (PARI) isok(k, n) = {for (x=0, n, if (gcd(fibonacci(k), k+x) == 1, return(0));); return(1);}
%o A299144 a(n) = {my(k=1); while (!isok(k,n), k++); k;} \\ _Michel Marcus_, Feb 05 2018
%Y A299144 Cf. A000045, A104714.
%K A299144 nonn
%O A299144 0,1
%A A299144 _Alex Ratushnyak_, Feb 03 2018
%E A299144 a(29)-a(36) from _Michael De Vlieger_, Feb 05 2018
%E A299144 a(37)-a(42) from _Jon E. Schoenfield_, Apr 24 2018